# Comparing sample variances of the same population

If s1 is the variance of a small sample, and s2 is the variance of a larger sample from the same population. Is s1 an unbiased estimator for s2?

I am thinking since a sample variance is an unbiased estimator of the population variance, then s1 and s2 are both unbiased estimators for sigma^2, so s1 and s2 should also be unbiased estimators for each other.

But all the observations in a sample are part of the population, yet the observations between samples doesn't necessarily include each other. Does this make their variances biased?

• Before going any further, you ought to read about the distinction between estimation and prediction. Here, $s_1$ predicts $s_2$; it cannot estimate it. – whuber Oct 24 '12 at 20:51
• We don't normally speak of 'estimators' for sample quantities but for population quantities. If $s_1$ is an unbiased estimate of its population variance, it's unbiased for$s_2$'s as well (since it's the same thing), as you say. You should clarify what precisely you mean by bias in relation to sample quantities. – Glen_b -Reinstate Monica Oct 24 '12 at 21:38
• @Glen_b I think bias here would be the difference of the expected value between s2 and s1. Since both E(s1) and E(s2) should be E(sigma^2), they are essentially the same thing as you said. – Saber CN Oct 24 '12 at 22:07
• @SaberCN If the quantity you're interested in is $\rm{E}(s_1)-\rm{E}(s_2)$, then that depends on what estimators you use to obtain them. If you used unbiased estimators, then the difference in expectations is zero. If you used MLEs or some other biased estimator then it isn't unless sample sizes are equal. – Glen_b -Reinstate Monica Oct 25 '12 at 1:12
• It's usually possible to use an unbiased estimate of the variance, yes. But the question didn't specify that the estimator used for for the two variances was necessarily an unbiased estimator, so I didn't assume it was. The second paragraph implied there that you were thinking about unbiased estimation, but not that it was the only possibility. – Glen_b -Reinstate Monica Oct 25 '12 at 6:22